steinmetz cp on the law of hysteresis part 3

In an alternating magnetic circuit in iron and other magnetic material, energy is converted inito heat by molecular magnetic friction.. e., the energy consumed bv hysteresis, mneasures a

A4 ,a erpresented at tfle Eleventh General Meetizg

of the Amee rican Institute of Electrical Eng-in

eers, Philadelfhlia, May i8th, 1894, President

Hcnston in the Choir '

AND THE

BY CHARLES PROTEUS STEINMETZ

CHAPTER I.-COEFFICIENT OF MOLECUTLAR MAGNETIC FRICTION

In two former papers, of January 19 and September 2T, 1892,

I have shown that the loss of energy by mnagnetichysteresis, due

to miolecular friction, can, with sufficientexactness, be expressed

bythe empirical

formula-:I = a B16whereH= loss of energy per cm3. and per cycle, in ergs,

B = amplitude of magnetic variation,

coefficient of molecularfriction,

the loss of energy by eddy currents can be expressedby

h _1N B2,where h = loss of energy per cm3 and percycle, in ergs,

z coefficient of eddy currents

Since then ithas been shown by lMr. R Arno of Turiin, that

the loss of energy by static dielectric hysteresis, i.e., the loss of

energy in adielectric inan electro-static field can be expressed

by the sameformula:

where R = loss of energy per cycle,

F = electro-static field intensity or initensityof dielectric

stress inthe material,

a = coefficient of dielectric hysteresis

Here the exponent 2 was found approximately to = 1.6 at

the low electro-static fieldintensities used

At the frequencies and electro-static field strengths met in

570

1894.] S'EINYMETZ ON HYSTERESIS 571

condensers used in alternate current circuits, I found the loss of

energyby dielectric hysteresis proportional to the square of the

Other observations madeafterwards agreed withthis result

With regardto magnetichysteresis,essentiallynew discoveries

572 STEINMETZ ON HYSTERESIS [May 18,

have not been mnade sinTce, and the explanation of this exponent

1.6 is still unknown

In the calculation of the core losses in dynamo electrical

ma-chinery and in transformers, the law of hysteresis has found its

applicationa, and so far as it is not obscured by the superposition

of eddy currents has been fullyconfirmied by practical

As anl instance is slhown in Fig 1, the observed core loss of a

high voltage 500 E w altornate current generator for power

transmissioni Thecurve is plotted with the core loss as abscisse

and the ter-minal volts as ordinates The observed values are

marked by crosses, while the curve of 1.6 power is shown by

the drawni line

The core loss is averylarge and in alternators like the present

machine, eveni the largest part of the total loss of ener,gy in the

machine

With regard to the numnerical values of the coefficient of

hysteresis, theobservations up to the time of my last paper cover

the range,

97Xj03=

Wrought iron,

Sheet iron and sheet steel ( 2.00 5.48 3.0 to 3.3

Soft cast steel and mitis metal 3.18 6.o 12 0

Hard cast steel 27.9

Welded steel 2 e I4.5 74.1

Cobalt "I.9

While no new materials lhave been investigated in the

mean-timue, for some, especially sheetiron and slheet steel, the range of

observed value of i has been greatly extended, and, I am glad

to state, mostly towards lower valuLe of -, that is, better iron

While atthe time of my former paper, the value of hysteresis

X 10' = 2.0, talen from Ewing's tests, was -unequaled, and

thebest material I could secure, a verysoft Norway iron, gave

d X l03- 2.275, now quite frequently vaiues, considerably

betterthan Ewing's soft iron wire are found, as the following

table shows, which gives the lowest and the highestvalues of

hysteretic loss observed in sheet iron and sheet steel, intended

for electrical maehiniery

1894.] STEINMETZ ON HYSTERESIS 573

The values are taken at random from the factory records of

the General Electric Company

1.94 1.94

As seen, all the values of the first column refer to iron

superior in itsqualityevenitothesample of Ewing^q X 10 2.0,

unequaledbefore

The lowest valuie is ^ X 10' = 1.24, thatis,38 per cent better

than Ewing's iron A sample of this iron I have here As you

see, it is very soft material Its chemical analysis does not show

anything special The chemical constitution of the next best

samnple j X 10 = 1.33 is almost exactly the same as the

con-stitution of samples C X 103 = 4.77and ^ X 103 - 3.22,

show-ing quite conclusively that the chemical constitution has no

direct influenice upon the hysteretic loss'.

In consequence of this extenision of § towards lowervalues,

the total range of C yet known in iron and steel is fromr

C X 101 = 1.24 in best sheet iron to q X 10( = 74.8 in

glass-hard steel, and a X 108 81.8 in manganese steel, giving a

ratio of 1 to 66

With regardto the exponenit X in

which I foundto beapproximnately = 1.6 over the whole range

of magnetization, Ewing has investigated its variation, and

found that it varies somnewhat at different magnetizationls, and

that its variation corresponds to the shape of the magnetization

curve, showing itsthree stages.'

1 J A Ewing, Philo8ophical Transaections of the Royal Society, London,

Juine 15, 1893.

574 STE]NMETZ 0N HYSTERESIS. [May 18,

Tests of the variationof the hysteretic loss per cvele as

fune-tion of the temperature have been published by Dr W Kunz',

for temnperatures from 20° and 800° Cent They show that with

rising temperature, the hysteretic loss decreases very greatly,

and this decrease consists of two parts, one part, whieh

disap-pears againi with the decrease of temiperature and is directly

pro-portional to the increase of temperature, thus making the

hyster-etic loss a linearfunction of the temperature, anid another part,

which has becomne permanent, anid seems to be due to a

perma-nent ehange of the m-olecular structure produced by heating

This latter part is in soft iron, proportional to the temperature

also, buitirregular insteel

CHAPTERII. MOLECULAR FRICTION AND MAGNETiC HYSTERESIS

In an alternating magnetic circuit in iron and other magnetic

material, energy is converted inito heat by molecular magnetic

friction The area of the hysteretic loop,with the AT MI F as

abscissse and the magnetization as ordinates, represents the

energy expended by the M I. F during the cyclic ehange of

magnetization

If energy is neitlher consumed nor applied outside of the

magnetic circuit by any other souLrce, the area of the hysteretic

loop, i e., the energy consumed bv hysteresis, mneasures and

representsthe energy wasted by molecular magnetic friction

In general, however, the energy expended bythe M M

F.-the area of F.-the hysteretic loop-needs not to be equal to the

molecularfriction Inthearmature of the dynamno machine,it

probably isnot, but, while the hysteretic loop more orless

col-lapses under the influence of mechanical vibrationi, the loss of

energy by molecular friction remains the sa-me, hence is no

longermeasured by the area of the hysteretic loop

Thusa sharp distinction isto be drawn between the

phenome-non of'magnetichysteresis, which represents the expenditure of

energy by the M M. F., and the molecuilar friction

In stationary alternating current apparatus, as ferric

induc-tances, hysteretic loss and inolecular magn-etic friction are

generally idenrtical

In revolving machinery,the discrepancy between molecular

friction and magnetic hysteresis may become very large, and the

magneticloop mayeven heoverturhred andrepresent, not

expen-1 eUtroteohni8che Zeitschrift, Arril5th, 1894.

1894.] STEINMETZ ON HYSTERESIS 575

diture, but production of electrical energy from meebanical

energy; or inversely, the magnetic loop may represent not only

the electrical energy converted into heat by molecular friction,

buLt also electrical energyconverted into mechanical miotion

Two such cases are showninFigs 2 and 3 and in Figs 4 and

Z In these cases the magnetic reluctance and thus the

indue-tance of the circuit was variable That is, the magnetic circuit

was opened and closed by the revolution of a shuttle-shaped

as function of the position. The curve a, couLnter E M F. or,

since the internal resistance is negligrible, the impressed E M F.

and curve M -_ magnetismn. If the impressed -E M F., E iS a

sine wave, the current c assumes a distorted wave shape, and

the produict of current anid E M. F_, W -C Erepresents the

energy. As seen, in this case t-e total energy is not equal to

-zero, i e., the a M F orself-induction E not wattless as usually

supposed, but represe-nts production of electr'ical energy in the

-first, conisumptlion inthe second case. Thus, if the apparatus is

driven by exterior power, it assumes thephase relation shown in

576 STEINMETZ ON HYSTERESIS [May 18,

Fig 2, arid yields electrical energy as a self-exciting alternate

current generator; if now the driving power is withdrawn it

drops into the phase relation shown in Fig 4, and then continues

to revolve and to yield mechanical energy as a synchronous

motor

The magnetic cycles or H-B curves, or rather for convenience,l

the C-A curves, are shown in Figs 3 and 5

As seen in Fig 5, the magnetic loop is greatly increased in

area and represents not only the energy consumed bymolecular

magnetic friction, but also the energy convertedinto mechaniical

power, while the loop in Fig 3 isoverturned or negative, thus

representingthe electrical energy produced, minusloss by

molee-ular friction

: X_

-~~~~~

FIG 3.

This is the same apparatus, of which two hysteretic loops

were shown in my last paper, an indicator-alternator of the

"hhummningbird" type

Thus magnetic hysteresis is notidentical with molecular

mag-neticfriction, butis one of the phenomrena caused by it

CHAPTER III.-THEORY AND CALCULATION OF FERRJIC

INDIUCTANCES

In the discussion of inductive circuits, generally the

assump-tion is made, that the circuit contains no iron Such non-ferric

inductances are, however, of little interest, since inductances are

almost always ironclad or ferric inductances,

1894.1 STEINMETZ ON HYSTERESIS 5

With our present knowledge of the alternating magnetic

cir-cuit, the ferric inductances can now be treated analyticallywith

the same exactnessand almost the same siimplicity as non-ferrie

inductances

Beforeentering into the discussion of ferric inductances, some

ternms will beintroduced, which are of great value in

simplify-ing the treatinent

Referrilig back to the continuous current circuit, it is known

that, if ina continu-ous current circuit a number of resistances)

Bradley 'PoXates Engrs, N.Y.' FIG 4.

ri, r2, 93 are connected in series, their joint resistance, R, is

the sum of the individual resistances:

If, however, a number of resistances, Ir r 3 r, are

con-nected in parallel, or in multiple, their joint resistance, R,

can-not be expressed in asimple form, but is:

Hence, in the latter case, it is preferable, instead of the tern

578 STEINMIETZ ON HYSTERESIS [May 18,

4resistance," to introduce its reciprocal, or inverse value, the

termi conduetanee" p = Theen we get:

"If a number of conlductanices, pn P2, p3 are connected

in parallel, their joined conductanceisthe sum of theinidividual

Hence the use of the termn "resistance" is preferable in the

case of series connection, the use of the reciprocal term

con-ductance," in parallel connection, and wehave thus:

"The joined resistance of a number of series connected

re-si ts ces is eqtal to thesum of the individual resistances, the

Joined conductance of anumberofparallel connected

conduct-ances is equal to the sum of the individual conductances."

In alternating current circuits, in place of the term

"resist-1894.] STEINMETZ ON HYSTERESIS 579

ance" we hiave the term "impedance,"' expressed in comnplex

quantitiesby the symbol:

with its two components, the "resistacie" r and the

"react-ae s, in the formula of Ohm's law:

E= C U.'

The resistance, r, givesthe coefficient of the E M. F in phase

with the current, or tlhe energy component of E M F., Cr; the

reactance, s, gives the coefficient of the E M F in quadrature

with thecurrent, or thewattless CoMponent of E M. F., Cs,botl

combined give the total E M F.

CW= C Vr +s2Thlisreactance, S, is positive asinductive reactance:

"'The joinied impedance of a numiiber of series connected

im-pedances, is the sum of the individual impedances, when

ex-pressed in complex quantities."

In graphical representation,impedances have not to be added,

but combined in their proper phase,bythe law ofparallelogram,

like the 1.M F.'S consumedby them

Thetermn '4impedance" becornesinconvenienlt, hiowever,when

dealinig with parallel connected circuits, or,in other words, when

several currents are produced by the same E.M.F.,in cases where

Ohm'slawis expressedinthe form:

It is preferable then, to introduce tlhe reciprocal of

"impe-1 "ComplexQuantities and their use in Electrical Engineering,'" a paper read

before Section A of the Initernational Electrical Congress at Chicago, 1893.

580 STElNAETZ ON HYSTERESIS [May 18,

dance," which may be called the "admittance" of the circuit:

F_1

As the reciprocal of the complex quantity

U = r -j8,the admittanee is a complex quantity also:

consisting of the component, p, which represents the coefficient

of current in phase with the E M F., or energy current, o E, in

the equation of Ohm's law:

and the component, CTwhich representsthe coefficient of current

in quadrature with the E M.F.,or wattless component of current,

arE

p may be called the "condcetance," a the "suseeptance" of

the eirculit Hence the conductance, p, is the energy component,

the susceptance, ?, the wattless component of the admnittance

anid the nLmerical value of admittaneeis:

v=

theresistance, r, is the energy component, the reactance, 8, the

wattless component of the impedance

U r-J 8rand the numerical value of impedance is

-e sl

u = t/r2 + 8'2.

As seen, the term " admittance " meansdissolving thecurrent

into two components,in phaseand in quadraturewith the E M.F,,

or the energycurrent and the wattless current; while the term

"' impedance" means dissolving the F.M.F.into twp coimponents,

in phase and in qluadrature with the curreint, or the energy

E M F. and the wattlessE M F.

It must be understood, however, that the "conductance" is,

not thereciprocal of theresistance, but dependsupon the

resist-ance aswell as upon the reactance. Only when the reactance

s - 0, or incontinuous current circuits,is the conductance the

reciprocal of resistance

Again, only in circuits with zero resistance =- 0,is the

sus-1894.] STEIN ETZ ON HYSTERESIS 581

ceptance the reciprocal of reactance; otherwise the susceptance

depends upon reactance and upon resistance

Fromthe definition of the admnittance:

By these equations, from resistaneeand reactanee,the

conduct-ane and susceptance can be calculated, and inversely.

Multiplying theequations forp and r, we get:

-:582 STEINMETZ 0N HYSTERESIS [May 18,

the absolute value of impedance,

u 4 r2 + S2

the absolute value of admittance

The sign of " admittance " is always opposite to that of

"im-pedance," that means, if the cuirrent lags behind the E M F.,

the E M F.leads the current, and inversely, as obvious

Thuswe can express Ohm's law in thetwo forms:

h' = U

and have

"TThe joined impedance of a number of series connected

im-pedlances isequal to the sum, of the individual impedances; the

joinedadmittance ofanumberofparallel connected admittances

is eqlual to the sumn of the individual admittances, if expressed

in complex quantities; in diagramm,natic representation,

com-bination by the parallelogram law takes the_placeof addition of

the complex quantities."

The resistance of an electric circuitis determined:

1 By direct comparison with a known resistance (Wheatstone

bridge method, etc.) This methodgives what may be called the

truie ohinicresistance of the circuit

2 By the ratio:

Volts consuLmed in circuit

Amperes in circuit

In an alternating current circuit, this methodgives not the

re-sistance, but the impedance

u/= V'r2+ s2

of the circuit

3 By the ratio:

Power consumed - (E M .)2

(current)2 Power consumed'

where, however, the "'power" and the "E M. F." do not

in-elude the work done by the circuit, and the counter E. M F.'S

representing it, as forinstance, the counter E N F. of amotor

In alternlating current circujits, this value of resistance is the

energy coefficient of the E. N F., and is:

r Eniergycomponent of E M F.

Total current

1894.] STEINMETZ ON HYSTERESIS 583

It is calledthe "equivalent resistanc" of the circuit, and the

energy coefficient of current:

- Energy comnponent of current

Total E, M F.

is called tlle "equivalent conductance" of the circuit

Inthe same way the valie:

8 = WattIess component of E M F.

Total current

is the "equivalent reactance," and

Wattless comnponent of current

Total E M F,

is the "equivalent8suceptance" of the circuit

While the true ohmic resistance represents the expenditnre of

energy asheat, inside of the electric conductor, by a current of

uniform deensity,the " equivalent resistance" representsthetotal

expenditure of energy

Since inan alternating current circuit in general, energy is

ex-pended nlot only in the conductor, but also outside thereof, by

hysteresis, secondary currents, etc., the equivalent resistance

fre-quently differsfrom the true ohmic resistanee, in such way as to

represent alarger expendituire of energy

In dealing with alternating current circuits, it is necessary,

therefore,to substituteeverywhere the values "equivalent

resist-ance," "equivalent reactance," "equivalent conductance,"

"equivalent susceptance," to iiiake the calculation applicable to

genaeral alternating current circuits, asferric inductance, etc.

While thetrue ohmic resistance is a conistant of the circuit,

depending upon the temuperature only, but not upon the E M F.e

etc., the "'equivalent resistance"- and "equivalent reactanee"

is ingeneral not a constant, but depends upon the E M F.,

cur-relt, etc.

This depenidence is the cause of most ofthe difficulties mret

in dealing analytically with alternatingcuLrrentcircuits containing

iron

The foremostsources of energy loss in alternating current

cir-cuits, oLutside of the true ohmic resistance loss, are:

1 Molecular friction, as:

(a) magnietic hysteresis;

(b) dielectric hysteresis

M84 STEINMETZ OJV HYSTERESIS [May 18,

2 Primary electric currents, as:

(a) leakage or escape of cuLrrent through the insulation,

brush discharge;

(b) eddy-eurrents in the conductor, or unequal current

distribution

3 Secondary or induced currents, as:

(a) eddy or Foucault currents in surrounding miagnetic

4 Induced electric charges, electro-static influence

While all these lossescan be included in the terms "1equivalent

-resistance," etc., only the magnetic hysteresis and the

eddy-cur-rents inthe iron will form the object of the present paper

I.-Alfaynetic IJsteresi.S.

To examinle this phenomenon, first a cireuit of very high

in-ductanee,but negligibletrue ohmic resistancemaybeconsidered,

thatis, a circuit entirely surrounded by iron; for iiistance, the

primary circuit of an alternatingcurrent transformer with open

secondary circuit

The wave ofcurrent produces in the iron an alternating

mag-netic flux, which induces in the electric eireuit all M F., the

,counter E M F. of self-induction If the ohmic resistance is

negligible,the counter E M F equals theimpressedE M.F., hence,

if the impressed . M F is a sine-wave, the counter E M F., and

therefore the magnetism whichinduces the counter F M F must

be sine-waves also The alternating wave of current is not a

sine-wave in this case, but is distorted by hysteresis It is

pos-sible, however, to plot the current wave in this case from the

hystereticeycle of magnetization

From thenumber ofturns n of theelectric circuit,theeffective

couniiter E M F. L andthe frequenley X of the current, the

max-imum magnetic flux M1 is found by the formula:

E= 4/2NitXX10;

hence:

M E4/2 7t 10Vfl N

1894.] STEINYMETZ ON HYSTERESIS 585

Maximum flux X1 anid magnetic cross-section Sgive the

max-31imumn magnetic induction B

If the miagnetic induction varies periodically between + B

and - B, the m M F varies between the corresponding values

+ Fand -F and describes a looped curve, the cycle of

hys-teresis

If the ordinates are given in linesofnmagnetic force, the

ab- r16,00 0 4- _ 14 00

Scissoe in tens of ampere-turns, the area of theloop equals the

energy consumed by hysteresis, in ergs per cycle

From the h-ysteretic loop isfoundtheinstantaneous value of

M.M F correspondingto aninstantaneous value of magneticflux,

that is of induced E M. F., and from the m M. F., F, in

ampere-tuLrns per unit lenigth of magnetic circuit, the length I of the

magnetic circuit, and the number of turns n ofthe electric

cir-cuit, are found the iiistantaneons values of current c

correspond-ing to a M M F. F, thatisa magnetic induction B anld thus

in-duiced E M F e, as:

n

586 STEINMETZ ON HYSTERESIS [May18,

In Fig 6 four magnetic cycles are plotted, with the maximumlh

values of magnetic inlductions: B = 2,000, 6,000, 10,000 and

16,000, and the corresponding maximuM M.M.F.'S: F= 1.8, 2.8,,

4.3, 20.0 They show the well-known h-ysteretic loop, which

be-conies pointed when magnetic saturation is approached

These magnetic cycles correspond to average good sheet iron

or sheet steel of hysteretic coefficient: 0033, aind aregiven

0 F B 2000

i1= 7i9 8X \ F 2.8

_1-W _.1 i Ct~~~~0-2.bl

Bradley Poates, Engr'8, N.Y.

with ampere-turnsper cmi as abscissoe and kilolinies of mnagnetic~

force as ordinates

In Figs. 7, 8, 9 and 10 the mnagnetism, or rather the magnetic,

induction, as derived from the i-nduced 'E M. F.1 is assumed as,

sine-curve For the ditfer-ent values of magnetic inductioni of'

this sine-curve, the corresponding values of m m. F.~hence of~

c-urrent, are taken from Fig. 6, a-ndplotted, givi-ng thius the

ex-cit'ing currenit required to produce the si-ne-wave ofmiagnetism;

1894.] STEINf2TETZ ON HYSTERESIS 587

that is, the wave of current, which a sine-wave of impressed

E M F will send through the circuit

As seen fromn Figs 4 to 10, these waves of alternating current

Fare not sine-waves, butare distorted by the superposition of

higlher harmonies, that is, are complex harmonic waves They

reach theirmaxinmum value atthe same tiimne with the maximum

of magnetism,thatis, 900ahead of the naximuminduced E.M.F.,

hence about 90° behind the maximum impressed E M F., but;

pass the zero line considerably ahead of the zero valule of

mag-netismu: 42, 52, 50 and 41 degreesrespectively

The general character of these curtrent waves is, that the

maax-imum point of thle wave coincides inL timne with the maximumn

point of the sine-wave of mlagnetism, but the current wave is

bulged out greatly at the risinlg; hollowed in at the decreasing

side Withincreasingmnagnetization, thle maxsimum of thecurrent

-388 STEINMETZ ON HYSTERESIS [May 18,

wave becomesmorepointed, as the curve of Fig.9,for B=10,000

shows, and at still higher saturation a peak is formed at the

max-imum point as in the curve of Fig 10, for B - 16,000. This

is the case, when the cuarve ofmagnetization reaches within the

range of magnetic saturation, since in the proximity of saturation

Bradley Poates, Engrls, NJ.

the current near the nmaximumpoint of magnetization has to rise

abnormally, to cause asmall increase of magnetization only

The distortion of the wave ofmagnetizing current is solarge

as shown here, onlvin an iron closed magnetic circuit expending

energy by hysteresis ornly, asin the ironclad transformeratopen

1894.] STEINMETZ ON HYSTERESIS M8

secondary circuit As soon as the circuitexpends energy in any

other way, as in resistance, orby mutualinductance, or ifan

air-gap is introduced in the magnetic circuit, the distortion of

the-current wave rapidly decreases and practically disappears, and

the current beconles moresinuLsoidal Thatis, while the

distort-ing component rem-ains the same, the sinusoidal component of

current greatlv increases, and obscures the distortion For

in-stance, in Figs 11 and 12 two waves are shown, corresponding

in mnagnetization to the curve of Fig 8, as the worst distorted

The curve in Fig 11 is the current wave of a transformer at 1

load Athigherload the distortion is still correspondingly less

The curve of Fig 12 is the exciting current ofa magnetic

cir-cuit, containing an air-gap, whose length equals .1 the lenigth

of the magnetic circuit These two curves are drawn in 3 the

size ofthe curve in Fig S As seen, both curves are practically

sine-waves

The distorted wave of current can be dissolved intwo

com-ponents: a true sine-wave of equal efective intensity andequal

powerwit1l the distorted wave, called the "equivalentsine-wave,"

and awattless hAiher harmonic, consistinig chiefly of a term of

trip]e frequiency

In Figs 7 to 12 are shown,in drawn linaes, the equ:ivalent

sine-waves,andthe wattlesscomplexhigherharmonics,whlich together

formri thedistorted current wave The equivalent sinle-wave of

M M F., or of ecurrent, in Figs 7to 10,leads the magnetism by

34, 44, 38 and 15.5 degrees respectively In Figs 11 and 12the

equivalent sine-wave almost coincides with the distorted curve,

and leads the magnetism by only 90,

It is interesting to note, that even in the oreatly distorted

curves of Figs 7 to 9 the maximum valne of the equivalent:

sine-wave is nearlythesame asthemaximuinvalue ofthe original

distorted wave of M m. ia., aslong as magnetic saturationiis not

approached, being 1.8, 2.9 and 4.2 respectively, agaimst l.8, 2.8

and 4.3 as inaximnuin values of the distorted curve Since by the

definitionthe effective valuLeof theequivalentsine-wave is the same

as that of the distorted wave, this meanis,that the distortedwave

of exciting current shares with the sine-wave the feature, that

themaximn-um valueandtheeffectivevalueh-iavetheratio: 4/2 + 1

Hence, below saturation, the inaxim-num value of tlhe distorted

curve can be calculated from the effectivevalue-wlmieh is given

by thereading ofan electro-dynamometer-by the same ratio as

:<590 STEIN1XETZ ON HYSTERESIS [May 18

with atrue sine-wave, and the mlagnetic characteristic can thus

be determined bymeans of alternating currents, by the

electro-dynamometer method, witlh su-fficient exactness.

In.Fig 13is shownthetruie magneticcharacteristic ofasample

of average good sheet iron, as found by the metthod of slow

-reversals by the magnetomueter, and for coi-nparisoir in dotted

-lines the saiine el-aracteristic, as determinedby alternating

cur.-rents, by the electro-dynai-nometer, with amnpere-t-urris per cm as

ordinates, and magnetic inductio-ns as abscissoe As seen, the

-two c,urves practically coin2ide9Bup10,000 to = ,14,000 1

1894.] STEINMIETZ ON HYSTERESIS 591

For higher saturations, the curves rapidly diverge, and the

electro-dynamnometer curve shows comparatively small M M. F.Ss

producing apparentlyvery high magnetizations

ThesaneFig.13 gives the curve ofhysteretic loss,in ergs per

cm.3 and cycle,asordinates, and magnetic inductions as abscisse

Sofaras currentstrengthandenergyconsumptionisconeerned,

the distorted wave can be replaced by the equivalent sine-wave,

and thehigher harimonies nleglected

All the measurements of alternating currents, with the only

exceptionof instantaneousreadings, yieldthe equivalent sine-wave

only, but snppress the higher harmonic, since allmneasuring

in-struments give either the mean square of the cuirrent wave, or

the mean product ofinistantaneous values ofcurrent and E M F

which are by definition the same inthe equivalent sine-wave as

in the distorted wave

Hence, in all practical applications, it ispermissible to neglect

the higher harmonic altogether, and replace the distorted wave

byitsequivalentsine-wave,keepingiuminid, however,theexistence

of a higher harmonic as apossible disturbing factor, wicth may

becomenoticeable in those very infrequLent cases, where the

fre-quencyofthe higher harinonicisnearthe frequencyofresonance

of the circuit

The equivalent sine-wave of exciting current leads the

sine-wave of magnetismn by an angle a, which is called the "angle of

Aysteretic advanceofphase." Hence the current lagsbehind the

E M F by 90 - a, and the power is, therefore:

P= CLecos (90°-a) = CEsina.

Thus the execiting current Cconsists of an energy comuponent:

Csin a, -which is calledthe "hy.3teretic energy current," and a

wattless component: Ccos a, which is called the "mnagnetizing

eurrent." Or inversely, the E M F. consists of an energy

com-ponent:: E sin a, the "hysteretic envergyE M F." anid a wattless

component: EGcos a, the E. m F of self-rnductton."

Denoting the absolute value of the impedance of the eircuit

by u-where u is determinedbythemagnetic characteristic of

theiron, and the shape ofthemagnletic and electric circuit-the

impedance isrepresented,in phase and intensity,bythe synmbolic,

expression:

U r-j s = u sin a-ju Cos a,

592 STEINMETZ ON HYSTERESIS LMay18.1

and the adimittance by:

C + os a = v sin a + IcosC a.

The quantities: u, r, sand v, p, a are not constanits, however,

in this case, as in the circuit without iron, but depend upon the

intensity ofmagnetization, B, that is, upon the E M. F

This dependence comnplicates thein-vestigation of circuits

con-taining iron

In a eirenit entirely enelosed by iron, ais quite considerable,

from 30 to 50degreesforvalues below saturation Hence even

with negligible true ohmic resistance no great lag can be

pro-dueed in ironclaClalternating current irculits

As I have proved, the loss of energy by hysteresis due to

molecularfriction is with sufficient exactness proportional to the

l.6th power of magnetic induction, B Hence, it can be

ex-pressed by the formula:

= the ccoefficient of hysteresis."

At the frequency, N, in the volume, VT, the loss of power is

by this formnula:

P = N FB-B' 10-7 watts,

- N V ( 1)0-7 watts,

where 8is the cross-section of the total magneticflux, Xl

The maximum magnletic flux, Mf, depends upon the counter

E M F of self-ind-uction, E, bythe equation:

E= V/27r Nn M 1O-8,

or,

iL=- E10where n = number of turnsof the electric circuit

Substitutingthis in the value of the'power, P, and cancelling,

we get:

= , N6 28 i-6 X 81.6 n'6- 58 N1.6 81.6 n,6